Simplify the following expression and state the condition under which the simplification is valid: $q = \dfrac{y^2 + 7y + 10}{y^2 + 2y}$
Solution: First factor the expressions in the numerator and denominator. $ \dfrac{y^2 + 7y + 10}{y^2 + 2y} = \dfrac{(y + 5)(y + 2)}{(y)(y + 2)} $ Notice that the term $(y + 2)$ appears in both the numerator and denominator. Dividing both the numerator and denominator by $(y + 2)$ gives: $q = \dfrac{y + 5}{y}$ Since we divided by $(y + 2)$, $y \neq -2$. $q = \dfrac{y + 5}{y}; \space y \neq -2$